Melt butter in a skillet over medium low heat; cook and stir orzo in melted butter until golden brown, 3 to 5 minutes. Stir in onion and cook, stirring occasionally, until translucent, 5 to 10 minutes. Stir in garlic until fragrant, about 1 minute.
Add broth and rice to orzo onion mixture; bring to a boil. Reduce heat to medium low, cover the skillet, and simmer until rice is tender and liquid is absorbed, 20 to 25 minutes.
Remove from heat and stir in spinach and feta. Cover and let stand until spinach is wilted and feta is melted, about 5 minutes. Fluff with a fork.
Sarah's Feta Rice Pilaf · Dinner Match Lab | Dinner Match Lab